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CREV REPORT 4343/84 CriDV RAPPORT 4343/84

a) FILE: 3621B 005 DOSSIER: 36218-005NOVEMBER 1984 NOVEMBRE 1984

I

ON THE INVERSION OF THE LIDAR EQUATION

B.T.N. Evans

Reproduced From JAN 24

Best Available Copy JN240

SIj

BUREAU - RECHERCHE ET D#VELOPPEMENT RESEARCH AND DEVELOPMENT BRANCH

MINISTiRE DE LA 'ODENSE NATIONALE DEPARTMENT OF NATIONAL DEFENCE

CANADA CANADA

NON C LASSIFi:Caad iDIFFUSION ILI.IMITZE

Canada ..8-5 01 14 095

DREV R-4343/34 UNCLASSIFIED CRDV W-4343/84FILE:' 3621B-005 DOSSIER: 3621B-005

II

ON THE-INVERSION OF THE LIDAR EQUATION

by

B.T.N. Evans

CENTRE DE RECHERCHES POUR LA D 9E

DEFENCE RESEARCH ESTABLISIHENT

VALCARTIER

Tel:' (418) 8444271

Qu~bec, Canada November/uoavebre 1984

NON CLAsint.

- I -s~i

UNCLASSIFIED

ABSTRACT

2 A modified form of the Klett inversion method of the lidar equa-

tion is developed. The accuracy of this inversion method. is explored

and validated. Unlike previous inversions, this one does not require

making a guess for each lidar return to initiate-the calculation.

Included are discussions of the other inversion methods. The

assumptions contained in the single-scatter lidar equatfon and in the

inversion method are outlined as is their relative importance. Special

attention is given to the backscatter versus total extinction relation,

multi-ocattering and non-ideal detector response.*- -

RESUME

On a modifif la ,mithode d'inversion d&velopp&e pa- Klett pour

l'Squation lidar. La validitf et l'exactitude de cette nouvelle

m6thode sont d6montrfes. Contrairement aux m~thodes dj&j connues, la

modification permet d'•viter l'emploi d'une estimation pour determiner

l'extinction i chaque retour du rayonnement lidar.

On discute aussi des autres m~thodes d'inversion. On souligne

1'importance des hypotheses dane le' d&veloppement de 1' quation lidar

pour ladiffusion unique et la amithode d'inversion. On accorde une

attention spfciale A la, relation entre la rftrodiffuoibn et i'extinc-!

tion totale, la diffusion multiple et I la r6ponse inadiquate'desd~tecteurs. jAccession For

"XT ISA

y e.. tion.

A.a.labillt7 Codes

.. A VAil and/or: •,,• 1 rlat [,Spoeta;l "

:....:.:,:.:;•..% .: ..,:.•..:.:- -, ...... , . .. ... . . .... . ..... .... ... .. ,...:.. . .. .. . ... ,.... ..... ... .... .....

S..•,.•,. -,•...,.,•.. .. •-.,, . ,. .. .... ,..,..•,.,.,.-,..• :. . ... :",:'; • ... ,,,.,,;. ' .-.\- .... ,. . .. .... .. -

UNCLASSIFIED

TABL OF CONTENTS

ASTRACr/RE3UME ...........-

NOMENCLATURE ... .. v*

1.0 INTRODUCTION 1

2.0 THE LIDAR EQUATION . . . . . . . . . 2..... . . . . . 2

3.o INVERSION OF THE LIDAR EQUATION . . . . . . . . . . . . . . . 4

3.1 Slope and Ratio.Method................. 53.2 Exact Forward Inversion Method i . . .........3.3 Exact Backward Inversion Method (Klett's Method) .7.. .. ,3.4 New Approach .(AGILE) .................. 8

4.0 VALIDATION OF AGILE . . . . . . . . . . . .......... 13

4.1 Remaining Problems 144.2 Tranmission Comparisons .............. . . 174.3 Concentration . . . . . . . ....... . 194.4 Clou.d Extent . . . . . . . ....... . 224.5 Clear Air Extinction ................. 22 ..4.6 Invariance of Inversion Point ............. 22

5.0 QUALITY OF ASSUMPTIONS ................... 23%,

5.1 Amplifier Response * • . ........... . ... 235.2 Multi-Scattering ........ ........ 245.3 Particle Size Distribution . . . . . . .... ... 27

6.0 SUMMARY AND CONCLUSIONS ................... 29

7.0 ACKNOWLEDGEMENTS ...................... 30

8.0 REFERENCES . . . . . . . . . ... . . . 31

FIGURES I to 11

TABLE I N

APPENDIX A - The Effects of Clear Air Calibration ...... 35

APPENDIX - Correction for the Logarithmic AmplifierResponse . 38 '

APPENDIX C - Numerical and Digitization Errors * • . .... 39

APPENDIX D - FORTRAN Program Listing of AGILE % . ..... . 43

_ _i

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NOMENCLATURE

A Receiver area jBV Planck emission function at frequency v

C(r) Clear air backscattered power, at distance r "

c Speed of light

c As - subscript refers to clear air

D Offset distance used in calculating digitization

and numerical errors

.d Constant of proportionality relating backscatter and extinction

F(r) System function including system sensitivity and geometric

crossover as a function of distance r

H Relative change between clear air signal and peak of signal

Iv Specific intensity of radiation of frequency v

i Jr. Source intensity at frequency V6.

Power law coefficiert relating backscatter and extinction'

L Incremental distance through an aerosol 6

a As a subscript refers to distance at which a guess is made

P(r) Lidar backscattered power return at distan e r

• . ,- -*-. -~. •-•-"•,,, * t,•~ , --.. ,I•.... -.... '.•........ .. " •-• - ..- ... .... .... ,.....; . ;•-• . ,•,• ..- : .,.,•- ,-• . • ..• .• -.. •.•" ,•,• ,-• • • • -.. •".• .. • . -. :..• -. ; -:, " .

UNCLASSIFIED

vi

o As a subscript refers to initial distance

p( p) Phase function as a function of the cosine of the scattering

angle p

r Distance or range from the lidar

S(r) Logarithmic range-adjusted power at range r

SE Experimentally determined S(r)

ST Theoretically determined S(r)

T Transmission

Tinv Transmission as calculated by the inversion

W Concentration

z Parameter for adjusting correction to logarithmic amplifier and

multi-scattering

a Mass extinction coefficient

O(r) Backb~atter extinction coefficient at range r

O(r) Voltme. extinction coefficient at range r

Laser pulse duration

Cosine of scattering angle

V Frequency

'-, . . ' , . ,

UNCLASSIFIED1

1.0 INTRODUCTION

The lidar equation is the governing relation which permits the

calculation of extinction and estimates of concentration under condi-

tions of s'ngle scattering from a given lidar return. However, to

obtain the solution, an inversion of the equation must be performed.,71 0. a-IAýVr ,"'r. I

Until lecently, the inversion of lidar returns to obtain extinc- e .

tion coefficients or concentration profiles has been plagued by insta-

bility and inaccuracies (Ref. 1). The instability of previous

solutions is found to be caused by mathematical and not, as some have

suggested, physical reasons. Inaccuracies can occ-r by marking unneces-

sary assumptions to further simplify the equaticn. The solution

proposed by Klett in 1981 (Ref. 1) goes a long way to improve this

situation; however, as will be shown in this report, it is not quite

ideal. In Ref. 2 Lentz describes a way to overcome some of these prob-

lems at high visibilities.

It seem that the inversion of the lidar equation (or radar

equation with attenuation) has not been givan the Attention it

deserves. Indeed, the basis of the ideas used in the recent solutions

was contained in a paper by Hitschfeld and Bordan in 1954 (Ref. 3).

This report is an attempt to eliminate or significantly reduce

the remaining problems while maintaining a simple algorithm. With the

inversion of the lidar equation reduced to a sixtle, efficient,

algorithm, analysis from many lidar, returns (e.g. from the Laser Cloud

Mapper (LCO), Ref. 4) is possible In a short time. Appendix D gives a

FOrTRAN IV listing of the final pregrm used in this report.

This work was performed at DREV between May 1981 and February

1983 under PCN 23305, Aeroeols.

711" .. •*

.

__*______,_.__.. .._.___..__. . . . •,.. . . e

UNCLASSIFIED

2

2.0 THE LIDAR EQUATION

To derive the lidar equation, it is convenient to begin with the

equation of radiative transfer (Ref. 5):

dlv V

dI 0 (r) j L-- -

where I, is the specific intensity of radiation of frequency v in the r

direction, a(r) is the total extinction coefficient and J is the

source term. J contains contributions to the intensity from scat-

tering or emission and can be given by

J'- B V + 41 p( 1) dI [21 p

where B v is the Planck emission function, p is the phase function fOr

scattering and p is the cosine of the scattering angle. Note that in

deriving [M] an important assumption has already been made: that the

distance between the scatterers is larger than the wavelength used. If

this condition holds, then the scattered radiation is incoherent and.

noninterfering.

For the next step in deriving the lidar equation, the. assumption

Is made that I. >> J r" This assumes that no significant amount of

emission or scattering from the medium are placed in the direction of

propagation. Note that this contains the critical assumption 'that only

single scattering events occur. In making this assumption we, obtain

1 dI v -d(r) [3

or

, . OVe "fo *(r')dr' [4]

.% % -. • %%

UNCLASSIFIED3

with I being the initial intensity of the beam before any extinction.

Equation . is of course the well-known Beer's law.

For light that is being received in the direction opposite' tc

the beam direction, the intensity received will be proportional to Iov,

geomtric attenuation A . where A is the receiver area, and the foldedr CT ,

laser pulse length given by -, where c is speed of light and T is the

time duration of the laser pulse. Furthermore, the intensity will

depend on the probability of being backscattered at a range

r, [O(r) - p(-l)], and finally a system function F(r) which includes

factors such as system sensitivity and geometric crossover of the

receiver and laser beam. Putting all these factors together with [4]

and noting the power received P(r) I (r), we obtain the lidar

equation:

CT F(r)O(r) Ae -2 frG(r')dr'Pelr) -Po -

where Po is the initial laser pulie power. Here, the extinction term

in Beer's law has been squared, since the returned power must tziverse

the same mediua twice.

In stmmary, the derivation of [5] has required the following

assumptions:

1. incoherent scatterers (Equation of Radiative Transfer);

2. no significant emission I, >> J" ;

3. first-order multiple scattering;

4. only backscatter is received, O(r) - p(-l) [1(r) could be

propoerly integrated over u]; oil

oI. -'--- .+,•.-....,-.,... .-... ,........,. . .. ,. :-?.:. .. :; .- • -:•.•..: ......-.... -.;,•, .: : • • ?• +.

• ."", '.:;• , '..' +".''" .'"..',..''.+." "• . " .:./ ..... ,.'-:',..\.-.: • - A-.•..-." .\.***' .•..' '- • • ,'• •

UNCLASSIFIED4

5. only a plane wave is received, i.e. even light distribution

over area A (one can use an effe-tive area);

6. no beam pulse stretching (T does nct change);

7. receiver encompasses laser beam; beau'overlap is complete

and there is no blind spot (1/r 2 );

8. invariance of the phase function over suall solid angles;

9. O(r) is independent of P o;

10. particles are not in the shadow of each other (particles act

independently); and

11. the light source is monochromatic.

3.0 INVERSION OF THE LIDAR EQUATION

In the' lidar equation [5], it -.s usually the tern 0(r) that is

the unknown of greatest interest. It can be seen however that this

equation has three unknowns: a(r), 0(r) and F(r).

F(r) is usually estimated or obtained by a calibration expert-

ment reducing the problem to two unknowns. In order to proceedi, a

further assumption must be made, that is

km

0(r) - dok [6]

Klett has made a literature search which indicates that

0.67 < k < 1,0 (Ref. 1). Indeed, it seems that for water fogs, k, I

is the value most often reported (with proportionality constant

d w 0.05 Sr- 1 (Ref.. 6, 7, 8)). However, for other types of aerosols

(e.g. snow, burning phosphorus clouds, silica dust, etc.), there is

W \i

•..-'.-..'. .. . . . .. ',n.*.*~.** ~ %%

UNCLASSIFIED5

less information available and furtaermore, if the size distributionchanges in regions involving the lidar signal, k will probably vary.

For a theoretical study, see Ref. 9. MNre will be said on this asstmp-

tion later in this report (Chapter 5.0).

Uith this final assumption [6], the lidar equation can now be

solved for a(r). In order to simplify [5] the following definition is

usually made:

S(r) - 2 P (r)] [7]

Taking the derivate of S(r) with respect to range, plus thelidar equation [5] with F(r) a constant and the .aswumption [6], we get

dS . kdd -2o [8d] odr a dr

In deriving [8], both F and d disappear. This will be important

In Section 3.4.

The following three sectitns briefly review the major inversion

methods to date and a fourth section describes the development of the

modified inversion method. All four sections discuss the strengths and

weaknesses of the various methods.

3.1 Slope and Ratio Method

Th.i.s method wili only be mentioned briefly since its value as a

solution has greatly diminished with the advent of the newer solu-

tions.

If one asstme that over some distance the aerosol is uniform,

we can have d-. O. Therefore, fra [8],.

d.

A.

UNCLASSIFIED6

1 dS rg

so that the slope of a plot of S versus r will give us the value of a

over the homogeneous region. This can be a very poor approximation if

there is noise or if the cloud is inhomogeneous. In general, an

aerosol that produces a positive slope in S (likely in a very

inhomogeneous medium) winl give negative o by this method!

The ratio method is just a variant of the slope method when tt

is applied segment by segment (see Ref. 10). Note also that [9] would

,result from the assumption that k - 0, i.e. the backacatter coefficient

is a constant independent of the cloud concentration. Even with the

problems listed above, it is the slope method that has been used the

most often in the past since it is simple and the only widely known

method.'

3.2 Exact Forward Inversion Method

In 1954, Hitschteld and Bordan (Ref. 3) solved the differential

equation [8] exactly. It is not difficult to solve because it is a'

standard differential equation, namely a homogeneous Ricatti equation.

The solution is

()exp[(S - So)/k]- [10)(o frexp [(S- So)/k]dr')

where the subscript o refers to the reference distance ro which to

closest to the lidar system. This solution requires knowledge of

ao which can be measured or guessed. However, as Mlett emphasized

(Ref. 1), this solution is unstable since two relatively large terms

are subtracted in the denominator -to obtain a small difference. Thus,

if there is an error in oa, the solution quickly becomes unstable and t

unrealistic values of a are computed at modestly small. r. See Refs. 1

J.-

- *1

* UNCLASSIFIED5° 7

and 2 for more details on the error analysis of this solution. It is

this instability that has caused the exact forward inversion method to

be neglected in favor of the slope method.

3.3 Exact Backward Inversion Method (llett's Method)

It seems surprising to find that, until 1981, no one had

published the idea of making the guess or measurement of a. in [10] at

the largest distance from the lidar system, even though this 'dea was

I mentioned in Ref. 3 and a more generalized version was discussed in

"Ref. 11. It Appears that some unsuccessful attempts were made

(Refs. 12, 13). However, using this idea, Klett obtained the following

- oolution:

exp [(S Sm)/k].:r) (al + k2 Irrm exp [ - S )/k]dr')

. where the subscript m refers to the reference dis.'ance rm which is

furthest from the lidar system. Now the two expressions are added in

"the denominator, making the solution more stable, and furthermore, the

error in a (which must be estimated for each return) becomes lessm

important as 'r decreases since the integral in [11] iucreases. Thus

the solution will tend to converge to the correct a(r) as r gets

smaller.

"I •Even though this solution is greatly superior to [10], some

problems still remain. If, for example, we take the case of high visi-

bility, it can be seen that the integral of [11 ] is very small and

Indeed can be considered near zero in comparison with a 1. If this

* .happens we get:

O(r) , am exp [(s S )/k]' 12

°

I

!

UNCLASSIFIED8

Thus all resulting extinction coefficients become linearly

related to a and hence to the error in am" From this, it is apparent

that, in the case of high visibility, the solution [ill is not very

stable.

This problem can be overcome (Ref. 2), by calibrating the lidar

system to find the system constants. Indeed, the very need for making

a guess (a.) in this solution is created by the derivation of the dif-

ferential equation [8] in which the absolute constants drop out. The

guess- can therefore be seen aa an attempt to put back this information.

Thus, in obtaining the system constant, one can make a better guess.

After a guess has been made, theoretically calculated values of S' (ST)

can be computed from estimated extinction coefficients. These, in

turn, can be compared with experimentally obtained S (S.) (computed

directly from the lidar return ,by SE - ln(P/C). The procedure in

Ref. 2 is to sum all ST and match them with SE by making

E(SR - ST) T 0 [131

The summation is intended to minimize noise in the signal. This method

produces an acceptable inversion for the high visibility case.

3.4 New Approach (AGILE)

A new approach called AGILE, (Automatically Generated Inversion

of the LIdar Equation) is developed in this section. This approach was

adopted because the following problems still remain with the Klett

method:

1. For the' high -visibility case, several iterations are needed

to minimize the sta in [13]. It is also possible that this

condition is not strong enough. For example, if, one has a

rapidly varying cloud of low density, thre will be' large

numerical errors occurring in the s8mantlo of [13]. These

UNCLASSIFIED9

errors could make the sum zero without it necessarily being

a good inversion (see the discussion of i,.merical errors in

Appendix C).

2. For the medium visibility case, the convergence of the Klett

method is still not fast ,enough, since perhaps half of the

values of the lidar return are spent converging to the

correct value. Therefore only the front half of the shot is

good. A procedure of the type used in Ref. 2 could help,

but will require on the average many iterations as well as

the evaluation of the integral fra(r')dr' with the ensuing

numerical errors.

3. At low visibilities, one can expect rapid convergence by the

Klett method, but now another problem art"ses which can make

the resulting inversion poor. This is caused by the theo-

retical limit of the integral in [11], as will be shown

below. If this limit is exceeded due to numerical or

experimental errors or incorrect assumptions, the resulting

inversion will give values for a that become increasingly

smaller than the true value as r decreases.

the proposed modification is an attempt to eliminate or signifl-

cantly reduce the above-mentioned problems in the inversion of a lidar

signal.

If we start with the lidar equation [5] and write it for the

case of the clear air returned power value C(r) (as done in Ref. 14),

we have

Cr -Ad oke -2acrCr P S r) c [14]r

UNCIASSIFIED10

where a is the clear air value of the extinction coefficient and we

I have taý7en a constant over r. If we now divide P(r) by C(r) and

rearrange, we get

-2a rk Pr)rr"e -2cr k P(r) rk e -215(r-)dr' [15]

e c

This result has assumed that the clear air aerosol and the cloud have

tile same value of d (which is not necessarily true but which does not

g affect the result). This is because we can take a c as an effective

clear air extinction coefficient (since we are not interested here in

"clear air c•tinction). Taking the k-th root of each side of [15] and

"integrating over r, one obta.nsI.'

1 2 a(r )r" 2 ar;(r')dr'- a [P-.-(- ke k C dr" - fo rO ) dr"

k[ 1-T(r) 2/kl [16]

where T(r) is the transmission at the distance r. To further simplify

"we will assume that ac is mall enough to be negligible (i.e.cdc << k/2r) to give

"-'2 hr(P) l/kdrw []. T2Ik] [17]

where it is now understood-that P, C and T deplne implicitly on r. The

importance of this result can be seeu when It is understood 'in terms of

its ,physical significance. Equation 17 states that the normalized

integrated backscatter has a limit. In dividing the power return P by

"" the reference signal C (in this •case, a' clear air shot), one calibrates

the laser shot against system constants, system noise and against the

i/r2 factor. The resulting improvement -if the signal is demonstrated

in Appendix A.I

UNCLASSIFIED11

The integral in [17] represents a sum over distance or time of

the power received during one shot or of the total power received (or

equivalently, the normalized integrated backscatter). Equation 17 then

states that this has a maximum, i.e.

2 r P/kcr (c ) dr" + 1 as T +0 [18]

k, d

If this limit is exceeded then there is either significant experimental

error (e.g. detector downtime), numerical/digitization errors or some

physical assumption is no longer valid (e.g. multi-scattering). This

then provides an immediate check, albeit weak, on the system, inversion

method and physical assumptions. It should be noted that limits simi-

lar to those in [18] can be derived for larger valves of r where a is

no longer small enough.

Another feature of [17] is that, knowing the sum of the power

received a't a given point, one can immediately calculate the transmis-

sion. This allows knowing T without doing the inversion to get a (and

consequently eliminates another pass through a numerical integration).

Finally, the integral in [17] is equivalent to the integral in

Klett's solution [11]. In fact, writing Klett'u solution [11] after

calibrating with C(r), one obtains

a c(P/C) 1/k

a (P /C,,/k II/C) /k [19]

_c.m m +!0 a mP dr'

The lidar equation can be written as (from [15])

a l (P/C)"/2ST 2/k (

%+

UNCLASSIFIED12

By noting that

ac (P /C)1/k 2 /k-TM [21]0 3

3

from [20], and

2ac Pm 1/k 2/k 2/kEfr ( dr'- (T T ) [22]

ko r / / /

"a generalization of [17],. it can be seen that Klett's slution is Just

"a restatement of the lidar equation (except that % or T are

guessed).

Again substituting [171 into [20], w get the solution in a

different form:

o - (P/C)/k d[23]-1 2 fr P Ilka• v ' c dr"

This is nimilar to the forward inversion method (Section 3.2) except

that 00 is replaced by a . Or, equivalently, Klett's solution becomes

0(P/c)i/k ,a fP(Q ik [24]

T 2/k /a + 2 Srm- dr

Equations 23 and 24 are equivalent since they guarantee that

sT(r) - s5(r) [25]

,at each interval of r, a condition much stronger thai [13s], providing

that errors do not invalidate (181. This can be seen because of [17]

and [20] and the definition of ST and S3, This condition, [25], Is

q'

4 .

Ta

- ,.. " . . "

iUNCLASSIFIED

13

not satisfied by Klett's solution [11] because of pocsible errors inthe guess, a With solutions [23] or [24], once a calibration shot

mhas been made and a value of a obtained, no guessing will be needed as

C'long as the value of o and the calibration, remain valid. (The values .

af c can be obtained by a trial inversion of one shot in which the

transmission is measured, or by other means). This has great advan-

tages over the other methods when many shots under the same conditions

must be processed. Plus, the necessary condition [25] is automatically

satisfied, whereas in the Klett method the weaker condition [13] must

be converged to and may still give significant errors.

4.0 VALIDATION OF AGILE

In this chapter, evidence of the success of AGILE will be

reviewed and compared with Klett's solution. The remaining diffi-

culties will be discussed, as will an indication of the numerical and

digitization errors. The lidar data used in this section was obtained

from the DREV Laser Cloud Mapper (LCK) (Ref. 4), with clouds 'produced

by burning id phosphorus.,

What constitutes a good inversion of the lidar equation [5]-

along with the ,assimption [6]?!

It is necessary and sufficient that a good inversion satisfy the

condition [25] and uses the correct value of ace Equation 13 'is only a

necessary condition.

Since many assumptions have been made to obtain [23] or [24],

how can a check be done to indicate that the results from the inversion,

are a good representation of the real values? The folloving item will

be compared and discussed in detail in this chapter in order to indi-

cat* the successes and limitations of the current inversion:

' 'I.. . . .. .. *,*..*.... . .. . . .

. . . . . . . . . . . . . . . . . .. . . .

UNCLASSIFIED14

1. the transmission of this inversion with respect to that

obtained from Klett's method (with the estimation procedure

as outlined in Ref. 2);

2. the transmission with that obtained directly by the LCM;

3. the extinction coefficients and concentrations found typical

by this method in burning red phosphorus smoke and those

from in situ measurements;

4.' cloud extent;

5. clear air extinction coefficients in front of and behind the

cloud; and

6. the results obtained should be independent of the Inversion

starting points in the shot.

4.1 Remaining Problems

With this. new approach, tvo calibration shots are needed: one

in clear air (or through a uniform sodium) and the other with some

transmission loss in order to determine ac" With the LCO this is not a

problem, since this ystem. is capable of making 100 shots per second,

all .in different directions and many through clear air. 9

Once a C and a good clear air shot have been obtained, the big-

lest remaining problem results from the possible errors in the evalua-tion of the integral in [23] or the integral and T In 2.This i

caused mainly by errors arising from the system resopose or multi-

scattering. For if the system response is too slow in recovery and/or

the cloud is dense enough for multi-scattering to be present, the limit e

[18] could easily be surpassed. If this happens, neither [23] nor [24]

16 ju

UNCLASSIFIED15

can work, and neither can the Klett solution. This occurs simply

because the lidar equation has become invalid.

If, however, the system response is not too bad, a correction

can be applied. Multi-scattering poses a much more serious problem and

there are currently no simple or accurate corrections for this phenome-

non. Therefore it is suggested that in performing an inversion using

the lidar equation [5], a guarantee of the limit [18] should be given.

This can be done by decreasing the return values in the individual

shots as the range increases, according to how rapidly the normalized

integrated backscatter *(or limit [18]) is approached. Since the func-

tional form needed for reducing the return values is not generally

known, estimates must be used (which adds some uncertainty to the

results). However, one, needs to know this function only in the extreme

case of very dense clouds and can be calibrated to som extent.

The functional form used for the multi-scattering is the same as

that used to correct for the logaritlhic amplifier (see Appendix B).

Thus a single correction term (or the sm form used by Ref. 15) with

one adjustable parameter attempts to correct for multi-scattering and

the amplifier response (and any other reason that may cause the limit

in [18] to be exceeded). This factor is just T-Z with z being the

adjustable parameter. Since both effects are combined, z will be

larger than if only one effect were present.

With the data presented In-this report, z ! 0.8 was found to

give good agreement with target reflectance. The value of this para-

mister In Ref. 15 is shown to be 0.2 < z < 0.3, with pure multi-

scattering, a field of view between 5 and 10 tarad and a size distribu-

tion similar to' that considered in the above-mentioned data. Thus for

the value of Z - 0.8, about 30Z is caused by multi-scattering. This

factor is used only when the optical depth becomes less than unity.

UNCLASSIFIED16

lee

Be

~ ea

z40

• 44

e i1 I , i w I n , I , , , I , , - "

2 -e l 86 "ee

" ,TeR iz OH F MI Il-ETT.

FIGURE 1. - Transmission comparison of ow inererion versus Kjett:'soJivtion

"2 -"", . .... . ..~. ... , ... ,,'"' -'

20

20

15 -.

0I10

- esponse of the logarthe amplfier

Z-TIe[ 110 NS)

%S

I-l I .'Io

UNCLASSIFIED17

0 0

0 0

0 0 DoATAo80t

00

80 - UNE OFEQUN.J1Y 0000

0 *% 0.

00.' 0

500

0 0

00z* 09 0us80 00i 0.0

200

*. rnst o 0omarion

0 0p

20 0

00

0 20 40 so so 100

FIGURE 3 -Comparison of transmissions by Inversion versus by targets

4.2 Transmission Comparisons

A comparison between the transmission computed from the present

method and that computed by Klett's method (Ref. 1) iB shown in Fig. 1.

The graph shows that there is excellent agresment between the two

methods. However, there is still aem scatter. One reason for dia-

crepancles is the difficulty in making the proper guess in the Klett

method in denser clouds, especially when there is significant signal at

the point in the shot where the guess is being made. This difficulty

is due to m erical and digitization errors that occur when calculating

the integrals and obtaining the transmission.

...

UNCLASSIFIED18

Another source of error is the response of the amplifier used in

processing the return signal. Figure 2 shows the response of the

logarithmic amplifier in the LI. The vertical scale is the loga--

rithmic output in arbitrary units, and the horizontal scale is the time'

in nanoseconds. From this, and with a digitization rate of 10 ns, one

can see that the rise time of this amplifier is adequate but that the

dcoutine will cause' problems, especially when the signal is large and

varies rapidly. Thus, in attempting to correct for this respocse (see

Appendix B or Section 4.1), errors will occur in both programs since

perfect correction is Impossible. 4

Figure 3 is a ploc of the transmission calculated from the new

algorithm versus the transmission as measured by the intensity of the

signal returned by cooperative targets. Again the agreeaent is reason-

able but there is some discrepancy. Most of this discrepancy can be

explained by the variation in target uniformity and errors in the laser r.

firing angle. These errors account for a variation of about 1 20%.

Other errors that can cause disagreement in Fig. 3 are the

problems outlined above, i.e. amplifier response, mmerical and digitit-

zation errors, plus, perhaps, significant multi-scattering for trans-

mission such less than 401 (about one optical depth). A least mean

squares fit of the poiats gives a straight line:

Ti - 0.91T1 + 5.18 [26]

with a correlasetl coefficient of 0.81, where TInv and TZ are the

transmissions obtained from inversion and experiment. respective-7. The

correlation coefficient value seems st reasonable when the variation

in the target reflectance Is considered.

%e 4" L

----- r\\j •

UNCLASSIFMD-19

4.3 Concentration

As already mentioned, the data here were taken by the LCM from a

red phosphorus generated smoke. By using values for the mass

extinction coefficients (taken from the DREV silo and = 50% relative.

humidity), a conversion of the total extinction coefficient into con-

centration is possible by the relation

[271

where a is the mass extinction coefficient, V is the concentration in

grans per cubic centimeter and L is the path length in centimeters.

This relationship has been verified by many studies (both theoretical

and experimental) over widely varying types of aerosols, densities and

wavelengths (see for example Refs. 16-19). With such a calculation of

concuntration, a comparison between the calculated and measured concen-

trations can be made. Unfortunately no direct measurement of the con-

centration was made when these lidar data were taken, but the purpose

here is to show that the concentration values obtained by the inversion

are reasonable "ballpark" figures.

DREV and others (Refs., 16, 20-22) indicate a - 1.5-2.5 m2 /g' at

1.06 ,tm and 50% R.H. Furthermore, one can obtain from the literature

In situ field values of the concentration for red phosphorus generated

smokes (Ref. 23). Table I gives a comparison between typical varia-

tions in concentrations between the calculation in this report and, in

Ref. 23. The correspondence between the two indicates a satisfactory

agreement. Thus the calculations dons here rive a good order, of magni-

tude estimate of the actual field values during the experiment.

" % . .

UNCLASSIFIED20

TABLE I

Comparison of field trial concentrations ofred phosphorus amoke

mg/2 3

Smoke Week III 5-200(Ref. 6)

Present Study 0.1-300

50

z 0Q

w 30

0_20

00

0 10 20 30 40

CLU WIDTH (M;)FIGUR 4 Comparison of cloud extent calculated by Inversion and by

photographic meaeur ee t.

N N0100 , ' , -*%*

,"'.'•'''J••-". . . 0• '..10 ,...;0 ,"30 40•'•ii .`•••••-[Jf••2F' 50`•• •`••'.'•,.. .-.', .,',..-'.,". ... .... .,••,.,•. .. . •.. .'.,-•CLO-U-..,,.D. -,-.,WI.. . -..TH .. .(". . ..).

•"~~IGR • o aio of clo.''ud.,5.-.;..'.•. ex,,n calclate .. byt..,,:_,.,2'' inerio an by.,,.' ,<'"..'., '"'"..• ... ','•.'.. :..".'photograph';' ! . i c• , .....a.s.u .. .r. .-e.,uen,.'t.,-".'. .,,...... ... ;.... .':".-.

, , , +

UNCLASSIFIED"21

*3089

I.ISO

0-1

15-2

t I a t,S5 9 ' I l , , , J , , , I S I I I I I, , I ,I , , ,, I

,40 6s o 8 1so 0 128 148 168 188

DTSTANCE I0D

• FIGURE 5 - Example of raw lidar return signal versus distance for an• RF smoke

!I

I- :

1 .-.s . I , , , i , . , I.. a I I _', , I i, .' I a , I

* OZSTNC I00

*' .FIGURE 6 -Inversion of the raw data Lu Fig. 5

"UNCLASSIFIED.22

4.4 Cloud Extent

During the trials, photographs were taken of both the front and

back of the cloud. This allows one to estimate the cloud extent photo-

4 graphically. Figure 4 is a plot of the calculated cloud extent along

one laser line and the estimate from the photographic data. Differ-

ences can obviously occur because of the difficulty in estimating the

location of the cloud edges. In the photograph, this is due to lack of

contrast and in the lidar data, it is due to the definition of what is

I cloud and what is clear air. The agreement is good. The Importance of

* this agreement will be discussed in the next chapter in relation to the

assumptions made.

4.5 Clear Air Extinction

Figures 5 and 6 are examples of a raw lidar signal and t'ie

*' resulting inversion for one shot. Figure 6 shows the calculated

extinction coefficient plotted against distance from the LCK (or lidar

-* system). The important feature to notice here is that the clear air

extinction values in front of and behind the cloud agree very well.

This is essential and verifies the calculated transmission through the

cloud.

4.6 Invariance of rnversion Point

- A necessary property that all Inversions should have -is that the

Inversion should be independent of the initial starting point, i.e.

overlapping results of differently initiated inversions should be iden-

tical. The new solution guarantees this because of the condition

SE T However for the Klett method, errors in the guessed value

will in general produce values different from guesses made at other

distances. Thus the resulting calculation' will be different. As the

cloud density increases and/or varies rapidly, this problem becoums

9

4

UNCLASSIFIED23

more important, eapecially for guesses made inside the cloud. This

invariance is obtained because of the normalization of the lidar return

by the clear air.

5.0 QUALITY OF ASSUMPTIONS

Many assumptions have been made in the ,derivation of [23] and

[24] and their subsequent use. An attempt is made here to identify the

most Important assumptions and their effect on the accuracy of the

inversion. Most of the assumptions leading to the lidar equation

(chapter 2.0) are excellent approximations to the conditions under

which the current data were taken. Major exceptions are the assumption

of single scattering (for dense clouds) and the invariance of the phase

function. In this case, it is a possible variation in the size dis-

tributions of the particles. This latter assumption is also involved

in 'the relation [6] and when k - 1, which is a constant with respect to

r. But perhaps the most important effect is that of the logarithmic

amplifier' s response.

5.1 Amplifier Response

The time resporse of the amplifier has already been given in

Fig. 2. A good check to see if 'the response of the amplifier is being

adjusted, for correctly (see Appendix B) is to pass a lidar return from

a target through the inversion with this correction. Figure 7 is the

result of such a calculation. Note that the program has now' considera-

bly compensated for the downtime and a relatively sharp response

remains.

The target response can be considerably enhanced with a Fourier

transform and deconvoluted with a Wiener filter (Ref. 24), but the

results, as indicated in Fig. 8, show thet the noise is considerably

amplified and the absolute values become unreliable.

." UNCLASSIFIED'

24

"Figure 9 is an illustration of results obtained from a scan of

"an HC (hexachloroethane) smoke without compensation for the logarithmic

amplifier. The disturbance in this cloud map occurs for extinction

"coefficients as large as 10-3 mn1 . Figure 10 shows the same cloud once

the logarithmic amplifier response has been corrected. Notice that

even at the lowest extinction coefficient, which is considerably

smaller than any in Fig. 9, there are no distortions of the cloud con-

touring. The' cloud maps of Figs. 9 and 10 are computer calculated and

"- " drawn and are in the form seen by the user of the program (Ref. 25).

Note that similar distortions due to multi-scattering could occur in

"clouds.

5.2 Multi-Scattering

The various experiments and theory that have so far been con-

"sidered in the literature make it very difficult to ascertain the

effects of multi-scattering for a given beam geometry, field of view,

lidar system distance, aerosol, wavelength, size distribution, etc.

"(Refs. 15, 26-28). Indeed, many of the parameters on which multi-

"scattering depend may only be known vaguely or not at all (e.g. parti-

cle size distribution and particle shape).

When the cloud is dense enough, significant multi-scattering

will invalidate the lidar equation. Thus, either a now lidar equation

"incorporating multi-scattering should be used. or the data adjusted so

that only the single-scatter component of the signal is considered.

By examination of the available theories and experiments (e.g.

Refs. 15, 26-28), a good& rule of thumb would be to consider multi-

scattering when the optical depth is greater than 0.1 for lidar systems

approximately I km or more from the cloud and perhaps for opticaldepths greater than 1.0 when the lidar system distance is close (about

50-100 i), as is the case with the LCM. This assumes that the field of

6 view of the lidar system is roughly in the range of I to 10 mrad."S..

-. * i** ' • J ¶) . . 2." '"'

UNCLASSIFIED25

1•e-tw'4

4~

I-I

S40 68 0 SeI 120 140

DXSTANCE (MD

FIGURE 7 Inversion of a target return

0.

'o I-

S 29 40 60 80

DZG=TZE 83N NUtBER

FIGURE 8 - Wiener filter as applied to the target return of Fig. 7

,r , ,'

UNCLASSIFIED26

V- asSKP.i ."M. e

.T L OW

+ Ir ... ,,,IVI•aIie..

11.44.4

1.0-4

FIGURE 9 - Lidar-produced cloud map without correction of logarithmicmplifier and multi-scattering

•'l-•!LiK1 I S OF SIG"A

.:=mff. totI FtiL+ PZLUC.UII t II111A, FILSOMMOSA$A.if"klII,61I I.l

II , lV f E IO II i

It ugmR iu.l . -+ l.e-i

C /,

311.t40N4 3.

,I si. I.i i,...

Wi.16 30.4

FIGUR 10 Sam cloud map as in Fig. , ith corrections

A%

UNCLASSIFIED27

The program that produced the examples for this report using the

new inversion method guarantees that the i.ntegral limit [18] Will not

be exceeded so that some compensation for imulti-scattering will be

present. No claim is made here that this compensation is complete or

correct.

5.3 Particle Size Distribution

It is difficult to imagine conditions in nonsolid particulate

clouds or smokes in which the particle size distribution does not

change. Varying humidities will affect the size distribution of hygro-

scopic particles for example. For' burning red phosphorus, the size

distribution will become stable only at some distance from the source,

even if the humidity is uniform. However, it was shown in Section 4.4

that there is good agreement between the calculation and the photo-

graphic evidence. Thus the value of k used cannot vary appreciably

from unity since this would change the cloud extent and hence bias

Fig. 4, since the inversion would give larger values. This can be

clearly seen inFigs. Ila and b, which show a plot of extinction

coefficient versus distance for various values of k. Values of k less

than unity affect the cloud extent to a greater degree than k > 1.

Notice that the extinction values are also changed, which would affect

the relationships already shown in Fig. 3 and Table I.

A survey of the literature indicates that for most conditions,

k - 1. Theoretical results (Refs. 7, 29-32) and experimental results

(Refs. 8, 17, 33-35) over a variety of materidls, particle shapes, and

extinction coefficients indicate that this assumption of linearity is

'excellent. However caution is required since, as indicated by results

in Refs. 8, 9, 33, 36-37, effects such as multi-scattering, changes of

size distribution (due to humidity or settling, etc.) and index of

refraction, and particle shape can cause nonlinearity. Values of

k - 0.66 have been reported (see Ref. 31' and the references contained

therein).

UNCLASSIFIED d28

I r0--2

Id

1.l-I .

let.

r. 6

146 146 1E 155 fIe 165 176 175

DZSTANc CFO

FIGURE Ila - Effect of powr law variation In the backscatter-extinction relation on Inversion results (k < 1)

'l C K SC A TTER-r T IC 1 1 H RELATIONsVARIr-O.4 OF POWER LAt J.

9-2

146 140 16 145 5O Is* la 17 17;

FI GUR lib - S al lie F 8. Il a but k 1 ,

UNCLASSIFIED29

As mentioned in Section 3.4, the clear air extinction coeffi-

cieat a can be interpreted as the effective extinction coefficient in

order to account for the differences between the proportionality con-

stant d in [6] relating the clear air aerosol to the aerosol under

study. The subsequent inversion using the calibration value of ac will

continue to be valid (under conditions discussed in Section 4.1) unless

the constant d in [6] for the material studied changes.

6.0 SUM4KARY AND CONCLUSIONS LI

In this report, a method has been developed that permits the

inversion of the lidar equation under a wide variety of conditions.

This solution overcomes instability in both the high and low visibility

cases, In contrast with the solutions proposed by Klett and Lentz

(Refs. 1, 2), by normalizing the signal by a clear air return and

considering malti-scattering. This solution is designed to invert the

lidar equation more accurately and with less computation.

It is demonstrated that there Is a theoretical limit to the

total power received (normalized integrated backscatter) in a return 4%

signal which provides:

1. a check an the validity of the lidar equation;

2. a check on the aystem itself;

3. transmission at any point, independent of the inversion;

4. that the transmission is obtained after only ons numerical

Integration instead of-two and Is therefore more accurate,

as a consequence of #3; and

- l

•'."..'.'•,-.t'.". " ".', . *-'•'-:'.•''.• •.•" .,-'• •'. "0"• •.**•'.'°. .. •.", *.• .

-. *- t-.~. • • :... ,.z • ' ,< .,• .- -% .. ,..' *• .. ,,- ..5 ....... "..:.. ,

UNCLASSIFIED30

5. a new inversion method without the need for a guess or a

need for iteration to yield convergence to some constraint.

The solution is validated In a number of ways, including:

1. comparison with Klett's method (Refs. I and 2);

2. comparisoa of predicted and measured transmissions;

3. comparison of calculated concentrations vith measured con-

centration values from the literature; and

4. cloud extent.

It is also shown that calibrating the lidar signal with a

reference signal (in this report, a clear air signal) makes the

r 2 correction unnecessary and considerably improves the ability to

detect a weak signal amongst significant system noise. As a result,

the values of extinction produced by the inversion are independent of

the starting point of the inversion.

7.0 AcnoLDGMETs

The author wishes to thank Mr. R.E. Kluchert for encouragement

and many helpful discussions, us well as Dr. W.G. Tam for help con-

corning the theory of mdti-scattering.

\,.

UNCLASSIFIED31

8.0 REFERENCES

1. Klett, J.D., "Stable Analytical Inversion Solution for ProcessingLidar Returns", Applied Optics, Vol. 20, No. 2, pp. 211-220,1981.

2. Lentz, W.J., "The Visioceiloeter: A Portable Visibility andCloud Ceiling Height Lidar", Atmospheric Sciences Lab., TR-0105,January 1982.

3. Hitschfeld, W. and Bordan, J., "Errors Inherent in the RadarMeasurement of Rainfall at Attenuating Wavelengths", J. ofMeteorology, Vol.. 11, pp. 58-67, 1954.

4. Evans, B.T.N.,'"Field Evaluation of a Canadian Laser Cloud Mapperand Candidate IR Screening Aerosols", Smoke/Obscurants SymposiumVI, Unclassified Section, Harry Diamond Labs., Adelphi, Md.,April 1982.

5. Chandrasekhar, S., "Radiative Transfer", Dover Pub. Inc.,Nev York, 1960.

6. Uthe, E.E., "Evaluation of the Propagation Environments using'Laser Radar Techniques", Office of the Director of DefenseResearch and Engineering Propagation Workshop, USAF Academy,Colorado, 5-9 December 1976.

7. Carrier, L.V., Cato, A. and Von Essen, K.J., "The Backscatteringand Extinction, of Visible and Infrared Radiation by Selected MajorCloud Models", J. of Applied Optics, Vol. 6, No. 7, pp. 1209-1216,1967.

8. Sztankay, Z.C., Mc~uire, D., Criffin, J., Hattery, W., Martin, C.and Wetzel, 'C., "Near-IR Extinction, Backscatter, andDepolarization of Smoke Week III Aerosols", Proceedings of theSuoke/Obsecurants Symposium V, Harry Diamond Labs., Adelphl, Md.,April 1981.

9. De Leeum, C., Mie Scattering on Particle Size Distributions.Influence of Size Limits and Complex Refractive Index on theCalculated Extinction and Backscatter Coefficients", PhysicsLaboratory TNO, PHL 1982-50, The Netherlands, 1982.

10. Kohl, R.H., "Discussion of the Interpretation ProblemEncountered in Single-Wavelength Lidar TransmIssometers', J. ofApplied Meteorology, Vol. 17, pp. 1034-1038, July 1978.

11. Fernald, F.C., Herman, B.M. and Reagan, J.A., "Determination ofAerosol Height Distributions by LIdar", J. of Applied Meteorology,Vol. 11, pp. 482-489, 1972.

4. - j.% ',

UNCLASSIFIED I32 i

12. Uthe, E., Private Comunication.

13. Carsvell, A., Private Communication.

14. Fujimura, S.F., Warren, R.E. and Lutomirski, i.F., "Lidars forSmoke, and Dust Cloud Diagnostics", Pacific-Sierra Research Corp.,Contract DAAK20-70-C-0040, May 1980.

15. Kunkel, I.E. and Weinman, J.A., "Monte Carlo Analysis of MultipleScattered Lidar Returns", Journal of the Atmospheric Sciences,Vol. 33, pp. 1772-1781, 1976.

16. Smith, R.B., 'Carswell, A.I., Houston, J.D., Pal, S.R. and Greiner,B.C., "Multiple Scattering Effects on Backscattering andPropagation of Infrared Laser Beams in Dense Military ScreeningClouds", Optech Inc. Final Report, Contract No. 8SD81-00084,February 1983.

17. Lamberts, C.W., "LIDAR. A Statistical Approach', PhysicsLaboratory, The Hague, Netherlands, PHL 1978-31, 1978.

18. Bruce, D., Bruce, C.W., Tee, Y.P., Chenxl, L. and Burket, H.,"Experimentally Determined Relationship between ExtinctionCoefficients and Liquid Water Content", Applied -Optics, Vol. 19,No. 19, pp. 3355-3360, 1980.

19. Pinnick, R.G., Jennings, S.C., Chylek, P. and AuvermnAn, H.J.,"Verification of a Linear Relation between iR Extlnction,Absorption and Liquid Water Content of Fogs", Journal of theAtmosphertc Sciences, Vol. 36, pp. 1577-1586, 1979.

20. Stuebing, E.W., "Relative eHmidity Dependence of the InfraresExtinction by Arosol Clouds of Phosphoric Acid", Proc. of theSmoke/Obscurants Symposium 11, Harry Dimond Labs., Adelphi, Md.,April 1979.

21. Allen, C., "Attenuation of Infrared Laser Radiation by RC, FS,WP, 'and Fog-Oil Smokes", Edgevood Arsenal,' Md., MR 405,. 1970.

22. Smith, N.D., Jones K. and Kittikul, P., "A Data ReductionTechnique for Field Data", Proc. of the Suoke/ObecurantsSymposium V, Harry Dimiond Labs., Adelphi, Md., pp. 97-125, 1981.

23. Farmer, W.N., Schwartz, F.A., Morris, R.D., Nornkohu, J.O. andBlnkley, M.A., "Field Characterization of Phosphorus Smokes usingSpectral Transmitttnce and Real-Time Concentration Measurements",Proc. SPIE-Int. Soc., Optical Engineering, Vol. 277, pp. 48-57,1981.

• - -. . , . ., ;.. .. . *... -. . . .

UNCLASSIFIED33

24. Boulter, JF., "Application of Deconvolution Filtering to Improvethe Range Resolution of a Ladar Target Imaging System", DREVTN-2120/74, March 1974, UNCLASSIFIED

25. Evans, B.T.N., Cerny, E. and Cagn&, L, "Computerized LidarDisplays of IR Obscuring Aerosols", Proc. Smoke/Obscurants off,Symposium VII, Harry Diamond Labs., Adelphi, Md., April 1983.

26. Tan, W.G., -Multiple Scattering Corrections for AtmosphericAerosol Extinctica Measurements', Applied Optics, Vol. 19, No. 13,pp. 2090-2092, 1980.

27. Pal, S.R. and Carswell, A.I., "Multiple Scattering in AtmosphericCloud: Lidar Observations", Applied Optics, Vol. 15, No. 8,pp. 1990-1995,,1976.

28. Samokhvalov, I.V., "Double-Scattering Approximation of LidarEquation for Inhomogeneous Atmospherew, Optics Letters, Vol. 4,No° 1, pp. 12-14, 1979.

29. Ross, D.A. And Kuriger, N.L., "Expected Lidar Return from TornadoClouds", Applied Optics, Vol. 15, No. 12, pp. 295&-29"), 1976.

30. Herrman, H., Kopp, F. and Werner, C., "Remote Measurements ofPlume Dispersion over Sea Surface using the DFVLR Minilidar",Optical Engineeripl, Vol. 20, No. 5, pp. 759-764, 1981.

31. Tvomey, S. and Howell, H.B., "The Relative Merit of White andMonochromatic Light for the Determination of Visibility by Back-Scattering Measurements", Applied Optics, Vol. 4, No. 4,pp. 501-506, 1965.

32. Bird, R.E., "Extinction and Backscattering by Fog and Smoke in the0.33 to 12.0 Micromter Wavelength Reg4on", IWC TR-5850, NavalWeapons Center, China Lake, Cal., 1976.

33. Waggoner, A.P., Ahlguist, N.C. and Charlson, R.J., "Measurement ofthe Aerosol Total Scatter-Backscatter Ratio", Applied Optics, Vol.11, No. ,12, pp. 2886-2889, 1972.

34. Kohl, R., Course on Atmospheric Optics Notes, University ofTennessee, Tullahoma, Tennessee, March 1982. -

35. Pinnick, R.G., Jennings, S.C., Chylek, P. and Ham, C.,"Ba'ckscatter and Extinction in Water Clouds%, Tenth InternationalLaser Radar Conference, Silver Spring, MD, 1980.

36. Sxtankay, Z.G., "Measurement of the Localized OpticalCharacteristics of Natural Aerosols, Smoke, and Dust", Proc. ofthe Smoke/Obscurants Symposium II, Harry Diamond Labs., Adelphi,Md.,. April 1978.

Iil l-•': -'•-'" . .. .. • -" J /S . . . t " " " - * . 1 I

UNCLASSIFIED34

37. Fenn, R.W., "Correlation between Atmospheric Backscattering andMeteorological Visual Range", Applied Optics, Vol. 5, No. 2,pp. 293-295, 1966.

i.-

I...

* * -.-. * *

.. ** ' .*. - * d** ~ ~**~' ~':--: 1

*..p-*~% .*.~>\'

UNCLASSIFIED35

"APPENDIX A

The Effects of Clear Air Calibration

In Section 3.4, where the new inversion method is developed, the

"lidar return signal is divided by a clear air return (or by any other

"reference signal). In so doing, the data are immediately corrected for

I/r 2 attenuation, the system constant F(r), and most system noise. In

this appendix, the sensitivity gained due to the latter effect will be

discussed.

TwO histogram are given in Figs. Al and A2 of the same lidar

ruturn showing the distribution, after inversion, of extinction coeffi-

cients. Here Fig. Al is the inversion w;:h just the 1/r 2 correction,

while Fig. A2 is the same inversion but using the clear air return for

. calibration. In both cases, the shot has been corrected for laser

power signal with respect to the calibrating shot. Also, the clear air

extinction coefficient was taken to be 2.0 x 10-5 m-1 . For this case,

the standard deviation 'is essentially halved by using the calibrating

shot. And, significantly, Fig. A2 allows for the detection of a very

weak return of the cloud while, from the statistics of the noise,

Fig. Al does not. Indeed, the one point in Fig. A2 (at 2.6 x .0-5 m- 1 )

"shows that the cloud is more than eight standard deviations from the

mean whereas it is only three standard deviations from the mean in

Fig. Al (at 2.3 x 10-5 m-l).

Figure A3 is the distribution for the extinction values of over

1200 clear air extinction coefficients. In this way, an excellent

, value of the standard deviation of the clear air extinction in the data

can be derived, enabling the detection of very weak returns. For

example, if an extinction value is over three standard deviations

• ...........

UNCLASSIFIED36

higher than the clear air median, the probability that the discrepancy

is due to noise, with the present data, is less than 0.27%. In this

way it is possible to distinguish between what is signal and what is

noise.

38

-J-

15,26

II14 IS Is 28 22 24 25 28

EX I 3NC I M(I COEF. Cl0*.6 1/11)

FIGURE Al Distribution of extinction- coefficients without clear airnormalization'

I,.

I UNCLASSIFIED37

•"~4 a 9 a I I I I 1 ! 1 ' 1 1 a I ''

CLU

I lie

"" 14 16 Is 20 22 24 25 28

A EXCTMON COEF. C106 1A0

FIGURE A2 - Distribution of extinction coefficients with clear airnormalization

I em , ' ' * I ' I '" ' I " * i ' ' ' I ' ' ' '

.. see

-5

460

L3in

2M6

4. '

14 MS Is 20 22 24 26 28

EMr3NC1IGH COEFV. CLj*.4 jAG0

FIGURE A3 Distribution of over 1200 extinction coefficients withclear air normalization

6

UNCLASSIFIED

38

APPENDIX B

Correction for the Logarithmic Amplifier Response

As shown in Fig. 2, the logarithmic amplifier response is far

"from ideal, and the use of digital filters can be difficult (as in the

one attempt discussed in Section 5.1). The method used in the current

inversion program to correct for this response is to multiply the

I return signal by a factor less than unity if the cloud is sufficiently

dense. This factor decreases continuously as the integrated back-

scatter gets larger. The actual factor used is TZ where T is the

V optical depth and z is the adjustable parameter.

The program halts if too much correction is required, assuming

that calculation beyond this point produces meaningless information.

"The amount of correction needed and when to halt the program must be

S empirically decided. This is done by comparison with transmission

values and cloud extent. Note that the amount adJusted for is con-

"trolled by a single parameter z and that this is made a constant.

A multi-parameter model was felt to be too flexible, i.e. it could be

I made to fit almost any condition. For the program used here, a value

of z - 0.8 was found to be reasonable; that i, it was a good fit with

'. the observed transmissions.

'S,

I,

9,

0S

UhCLASSIFIED39

APPENDIX C

Numerical and Digitization Errors

In calculating the inversion, a numerical integration must be

performed, with its inherent errors. Furthermore, the digitization of

the analog signal obtained by the LQ( also places an unavoidable error

in the data, which is due to the finite word size of the digitizer.

The purpose of this appendix is to estimate the importance of these t

errors. The errors produced in the integration and digitization of t..e

artificial signal shown in Fig. Cl will be discussed. This is mathe-

matically a simple representatioa of a range-corrected lidar return

signal, with the height H and an offset D being adjustable. The para-

meter H is taken as a relative change between the clear air, signal and

the plateau height and D is in units of the digitization rate.

The error will be taken here as the difference between the theo-

retical area under the curve (to the first data bin after the signal

height H) and the numerically derived area.

i • "SIGNAL

DIGITIZED /H, (HE]GHT)

81IN LOCATION! i

SAIR*, |

* .

(DISPLACEMENT)

FIGUIE Cl Artificial lidar return used for estimating numerical anddigitization errors

UNCLASSIFIED40

TABLE CI

Variation of Integration Error with Offset' andSignal Height (Infinite Word Size)

Signal Height (H)

1 10 100 1000

0 0 <0.5 <0.5 <0.5OFFSET (D) 0.1 0 3 8 10

0.5 0 13 55 900.9 0 30 150 470

TABLE CII

Variation of Digitization Error with Offset and."Signal Height (8-Bit Word Size)

Signal Height (R)

1 10 100 1000

0 0 1 50 90OFFSET (D) 0.1 0 20 80 110

0.5 0 30 120 3000.9 0 40 170 800

TABLE CIII

Variation of Digitization Error uLth Offset aedSignal Height (16-Bit Word Size)

Signal Height (R)

1 10 100 1000

0 0 0.8 8 10OFFSET (D) 0.1 0 13 40 70

0.5 0 20 90 2000.9 0 30 150 600

S:.'-.'. ." .: . * . . * - . .p

41

C1.1 Integration Errors

Both the offset D and height H of a "Idar return can vary and,

in the case of a dense make, H can vary considerably even within the

I best spatial resolution available to the LWI (1.5 a). For the program

of this report, Simpson's Rule sas used since it is simple to apply and

the theoretical error term Is of the fourth order in the digitization

rate. This is the most accurate method for numerical integration under

conditions of a uniform digitization rate and at the same time gives an

integrated value (in combination with the trapezoidal rule),at every

digitized point. Table CI lists the errors associated with the numerl-

cal integration of Fig. Cl for ranges of offset from 0-0.9, signal

heights of 1 to 1000 and an infinite digitization word size. It is

evident that large offsets can be tolerated only if the signal remains

relatively mall.

C1.2 Digitization Errors

The digitization errors have two sources other than that which

is included in the nmerical integration error. These are the finite

word length and round-off error. In the calculation done here, the two

I are treated together. Tables CII and CIII are the errors obtained by

varying the word size, signal height H, and the offset D. It can be

seen that for a certain combination of H and D, there is a small range

of word sizes in which the error increases rapidly.!

I

I

UNCLASSIFIED

42

C2.0 DISCUSSION OF ERRORS

The LOI digitizes lidar data with an 8-bit word size. In addi-

tion, the maximum observed relative change in signal strength due to

red phosphorus or HC smokes is 20 times over 1.5 m. Assuming, as a

rough approximation, that the signal increases linearly in the inter-

val, the maximum error due to numerical integration occurs for an off-

set of 0.5 when the signal has increased 10 tines. Table CI then gives

a' value of 13Z for the error due to numerical integration. For the

same case, the digitization error becomes 15Z. In general, the rela-

tive change in the signal is considerably less than 20 times and errors

less than 5-10% seem to be more likely. The Integration over the total

asignal will have less error due to less signal variation and same error

cancellation.

It is clear from the above-mentioned Information that computa-

,tional errors depend strongly on the signal variation. Therefore the

inversion method presented in this report incurs less error than the

other methods since transmission can be derived from one integral (the

normalized integrated backascatter) instead of two (i.e. the normalized

integrated backscatter and the integral of extinction values).

. .%I .

UNCLASSIFIED43

APPENDIX D

FORTRAN Program Listing of AGILE

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UNCLASSIFIED44

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UNCLASSIFIED45

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UNCLASSIFIED50

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